Structural Folding and Multi-Highest-Weight Subcrystals of $B(\infty )$

Abstract

We introduce a procedure to fold the structure of a crystal B of simply-laced Cartan type \({\mathscr{C}}\) by the action of an automorphism σ. This produces a crystal Bσ for the folded Langlands dual datum \({\mathscr{C}}^{\sigma \vee }\) which properly contains the well-studied \({\mathscr{C}}^{\sigma \vee }\) crystal of σ-invariant points. Our construction preserves normality and the Weyl group action, and is compatible with Kashiwara’s tensor product rule. Combinatorial properties of \(B(\infty )_{\sigma }\) reflect the structure of a subalgebra of \(U_{q}^{-}({\mathscr{C}})\), which is naturally a module over the graded-σ-fixed-point subalgebra of \(U_{q}^{-}({\mathscr{C}})\) via Berenstein and Greenstein’s quantum folding procedure. We find that \(B(\infty )_{\sigma }\) is generated by a set of highest-weight elements over the monoid of root operators. Through the Kashiwara-Nakashima-Zelevinsky polyhedral realization, the highest-weight set identifies with a commutative monoid which admits a Hilbert basis in finite type. A subset of the Weyl group called the balanced parabolic quotient is in one-to-one correspondence with the Hilbert basis for the pair \(\left ({\mathscr{C}}, {\mathscr{C}}^{\sigma \vee } \right ) = \left (D_{r}, C_{r-1} \right )\), and identifies with a proper subset of the Hilbert basis in other finite types. We obtain an explicit combinatorial description of the highest-weight set of \(B(\infty )_{\sigma }\) by establishing a connection between the action of root operators on \(B(\infty )\) and the semigroup structure in the polyhedral realization.